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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

0
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1answer
27 views

Proving a limit converges to -3 using definition of convergence.

so I have the problem $\lim\limits_{n \to oo} \frac{2-3n^2}{n^2+2n+1}$. I have to prove this using the epsilon definition. So I know the limit equals -3. So I do |$\frac{2-3n^2}{n^2+2n+1}$ + 3 | < $...
1
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0answers
20 views

A problem from Folland: constructing Haar measure from Lebesgue measure

The following is a result from A Course in Abstract Harmonic Analysis by G.B. Folland. It also appears as an exercise in the real analysis text by the same author. The context is Haar measure on a ...
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0answers
25 views

Prove that any bounded subset of $\mathbb R^2$ is totally bounded.

(Note I am seeking a proof before the development of the definition of compact sets; preferably one that finds an open cover of the subset.) My attempt: I tried to connect it to a proof developed ...
2
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1answer
33 views

How to show that $f(x)$ is not continuous using open sets?

I am practicing topology with a book, and I'm trying to understand the following definition. A map $f$: $\mathbb{R} \rightarrow \mathbb{R}$ is said to be continuous if for every open subset $U \...
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0answers
31 views

Prove by constructing bijective functions that the following sets $A$ and $B$ have the same cardinality. [on hold]

Prove by constructing bijective functions that the following sets $A$ and $B$ have the same cardinality: Let $A=\mathbb{N}$, $B=\{\{x_n\}^{\infty}_{n=1}: x_n \in \mathbb{N}\cup \{0\} \hspace{0.5cm} \...
1
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0answers
17 views

Cantor-Like Sets

I'm working on a problem from Stein and Shakarchi's Real Analysis, and it asks to construct a measurable subset $E\subset [0,1]$ such that for any non-empty sub-interval $I$ in $[0,1]$, both $E\cap I$ ...
0
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0answers
12 views

Minorization Condition for a Truncated Normal Distribution

Suppose $$t_1, \ldots, t_n | \beta, \sigma^2, y \sim \text{TN}(\lambda(y_i-x_i^T\beta),\sigma^2;0;\infty),$$ where $\lambda \in \mathbb{R},$ and $x_1,\ldots, x_n$ and $y_1,\ldots,y_n$ come from the ...
1
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1answer
52 views

Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$

Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$. Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
3
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4answers
53 views

Show that F vanishes at infinity.

Suppose $1 ≤ p < ∞, f ∈ L^p(R)$, and $F(x) = \int_{x}^{x+1} f(t) dm(t)$ Prove that F vanishes at infinity. We know that $\int_R |f|^p < \infty$, then, of course, for any $x, F(x)< \int_x^{...
6
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0answers
54 views

If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense?

If $X$ is a complex Banach space, is the set $T \in L(X)$ with finite dimensional kernel dense? Here $L(X)$ is the set of bounded linear operators from $X$ to itself equipped with the norm topology. ...
0
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0answers
31 views

Cantor's Theorem proof

I'm having problems understanding the Cantor's theorem proof. Can't we just assume $f$ is surjective, say the cardinality of $P(A)$ is greater than $A$ so by the pigeonhole principle there must be an ...
1
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2answers
35 views

Proving that $y_n$ converges given $y_{2n}, y_{2n+1}$ converges

Suppose we have $\mathrm{lim}_{n \rightarrow \infty} y_{2n} = \mathrm{lim}_{n \rightarrow \infty} y_{2n+1} = M \in \mathbb R$. I'm trying to prove from this that $\mathrm{lim}_{n \rightarrow \infty} ...
7
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2answers
58 views

Proving a specific set in $\mathbb R^2$ is closed.

I am trying to prove that the set $A=\{(x,y)|x^2\leq y\}$ is closed in $\mathbb R^2$. I wrote a proof, but I think the end is wrong. My proof is: Consider the set $A=\{(x,y)|x^2\leq y\}$ in $\mathbb ...
0
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4answers
68 views

When are we not allowed to replaced $\sin{x} $ , $\tan{x}$… by $x$ in a limit where $x\to 0$?

In what situation, can we not replace for example $e^x$ with $x$ when when $x\to 0$, in a limit. Sorry if the question is extremely vague , English is not my native language. Thank you in advance
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3answers
90 views

Can someone help me solve this limit? [on hold]

Can someone help me solve this limit? Thank you in advance. $$\lim_{x\to 0} \bigg( \frac{1+\tan{x}}{1+\sin{x}} \bigg)^{\frac{1}{\sin{x}}}$$ If possible without L'Hospital, the exercise gives more ...
0
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0answers
20 views

If $f:[a,b]\to\mathbb{R}^2$ ($n>1$) is a continuous rectifiable path, then m(f([a,b]))=0 [duplicate]

If $f:[a,b]\to\mathbb{R}^2$ is a continuous rectifiable path, prove that the measure of $f([a,b])$ is null. I've tried to use the fact that $f$ is uniform continuous, but I only got that $m(f([a,b]...
0
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0answers
29 views

The set of bounded Linear operators $B(V)$. Is $B(V)$ bounded?

I have a question, regarding the set of bounded linear operators. I was wondering is that set itself bounded? Furthermore if it is not bounded then what restrictions can we put onto the vector space $...
0
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0answers
16 views

Understanding the proof of inversion formula for density using characteristic function

The formula is: $f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda x}\hat{f}(\lambda)d\lambda$ where $\hat{f}$ is the characteristic function, $f$ is continuous bounded on $R$ and both $f,...
3
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2answers
57 views

Integrable function of which the antiderivative is not equal to the integral

Is there a Riemann-integrable function $f: [a,b]\to\mathbb{R}$ such that $f$ has an antiderivative which is not equal to $t \mapsto \int_a^t f(x)dx +c$ for any $c\in\mathbb{R}$?
1
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1answer
50 views

Hardy-Littlewood Inequality for Sobolev spaces

After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm. Fix dimension to be $3$. H-L-S says that ...
24
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7answers
4k views

Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero?

If $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq ...$$ and has the property that $\space$$a_{n+1}-a_n \longrightarrow 0$, Then can we conclude that $a_n$ is convergent? $$\space$$ I ...
1
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0answers
53 views

Continuous $f:(a,b)\to \mathbb{R}$ is convex iff $x \mapsto f(x)+\gamma x+\delta$ does not attain max for all $\gamma,~\delta\in \mathbb{R}$

Let $f:(a,b)\to \mathbb{R}$ continuous. Prove that $f$ is convex iff for all $\gamma,~\delta\in \mathbb{R}$ function $x \mapsto f(x)+\gamma x+\delta$ does not attain its maximum value on $(a,b)$. ...
1
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0answers
13 views

Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ ...
1
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1answer
42 views

(Proof verification) Showing that the two $\limsup$ definitions are equivalent

I have been trying to prove that the two definitions of $\limsup$ are equivalent. I would appreciate it if someone could verify my attempt! Thanks in advance! Here are the two definitions: ...
1
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1answer
16 views

Cover $\lambda_n$-null set with small cubes

I'm reading proof that set with $\lambda_n$ measure zero have also $\mathscr{H}^n$ (Hausdorff) measure zero. And there is claim: If $n\geq 2$, for every given $\epsilon,\delta > 0$ and Borel $\...
1
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1answer
47 views

A difficult limit -> problem in getting out of the form $\infty \times 0$

Let $(x_n)$ be a real sequence which converges to $l$. Moreover we have : $\mid x_n -l\mid \leq C \mid x_{n-1}-l\mid, C \in ]0,1[$ and : $\lim_{n \to \infty} \frac{x_{n+1}-l}{x_n-l} = k \in ]0,1[$. ...
0
votes
1answer
30 views

Convergence of certain integrals to $f(t)/2$

Let $f \in {L^1}\left( {0,T} \right)$. Prove that for a.e $t \in \left( {0,T} \right)$ we have $$\mathop {\lim }\limits_{n \to + \infty } n\int\limits_t^{t + \frac{1}{n}} {\left[ {1 - n\left( {s - t}...
0
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1answer
23 views

Bijectiveness in a neighborhood of $(0,\pi/2)$

Question: Define $g: \mathbb{R^2}\to \mathbb{R^2}$ by $g(x,y)=(y\cos x,(x+y)\sin y)$. Show that g maps a neighborhood of $(0,\pi/2)$ bijectively to a neighborhood of $(\pi/2,\pi/2)$. What I did/...
-1
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0answers
9 views

how functional reduce to Y(x,y,y') = y(x) + e x n(x)

How this functional can be expressed as above linear function . what math concepts and topics clear this fully and meaningful mathematical explanation.
4
votes
1answer
42 views

Trying to understand a standard example of weakly but not strongly measurable function

I consider quite a standard example of a function that is weakly measurable but is not strongly measurable. Unfortunatelly, I don't fully understand it. Let $X=l_2([0,1])$. Then $X$ equipped with a ...
0
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1answer
36 views

What is the function you get when you double the arguments of sin and cos in the Fourier Series of another function?

Suppose $$f(x) =\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(kx)+b_k\sin(kx) $$ Then what is $$S(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k\cos(2kx)+b_k\sin(2kx)$$ in terms of $f(x)$. I tried writing down ...
1
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0answers
44 views

Generalized Hermite interpolation

I'm trying to understand this topic, because I found it in an exercise of Zorich - Mathematical Analysis I. So, I have $p$ points $x_1,...,x_p$ and I want to find a polynomial $H(x)$ such that: $f(...
1
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1answer
20 views

second order homogeneous differential equation method

I'm trying to understand the general method to solve the second order differential equations $u''+2au'+bu=f(t)$. The homogeneous differential equation $u''+2au'+bu=0$ can be solved looking for a ...
0
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2answers
21 views

Example of a function which is continuous and $1$-periodic but not differentiable

Give an example of a function $f:\Bbb R \to \Bbb R$ such that $f$ is continuous and periodic with period $1$ but not differentible My try Consider the function $f$ whose graph is Actually it is a ...
0
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1answer
34 views

Preimage of continuous one-to-one function on connected domain is not continuous.

I know that given $B$, a compact subset of $\mathbb{R}^n$, and $f : B \to \mathbb{R}^m$, a continuous injective (one-to-one) function, $f^{-1}$ is continuous on $f(B)$. (This true). I also know that ...
2
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1answer
50 views

Exercise 36 Ch 1 in Stein's Real Analysis [duplicate]

Construct a measurable set $E\subset [0,1]$ such that for any non-empty open sub-interval $I$ in $[0,1]$, both sets $E\cap I$ and $E^c\cap I$ have positive measure. [Stein's Hint: For the first ...
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0answers
26 views

Discussing the convergence of a sequence $x_n$ [on hold]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
2
votes
1answer
22 views

Extension of domain of contactive semigroups, and why contractivity is important?

I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this ...
1
vote
1answer
42 views

I am stuck somewhere in this analysis problem

Consider a real valued continuous function f on [0,1] such that f is differentiable on (0,1) and f(0)=f(1)=0. Does there exist some c in (0,1) where f(c)=f'(c)? It seems like the answer is yes. I am ...
3
votes
0answers
46 views

Proof that continuous curve in $\mathbb{R}^2$ has Lebesgue measure zero

Suppose $\Gamma$ is a curve $y = f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0.$ [Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$.] My attempt: ...
3
votes
1answer
101 views

Prove that inequality $\sqrt{2\sqrt{4\sqrt{8…\sqrt{2^n}}}} \leqslant n+1$

Let $n$ be the integer. Prove that $$\sqrt{2\sqrt{4\sqrt{8....\sqrt{2^n}}}} \leqslant n+1$$ SOURCE: BANGLADESH MATH OLYMPIAD I am a new beginner at the infinite radical and sequence. I don't know ...
0
votes
1answer
45 views

An analytic doubt from Riemann zeta function-Titchmarsh

Doubt form the Book Theory of Riemann Zeta Function by Titchmarsh I've problem on Theorem $5.8$ in the equation $5.8.2$. When $k=l$ the from Lemma $5.7$ we get, $$ \sum_{n=N+1}^bn^{-it}=O\left(N^{1-1/...
1
vote
1answer
41 views

Proving $\lim_{x\to x_{0}} f(x) = 0$ implies $\lim_{x\to x_{0}} f(x)g(x) = 0$ in $\mathbb{R}^{n}$

Let $A$ be a subset of $\mathbb{R}^{n}$ and let $x_{\star} \in \mathbb{R}^{n}$ Also, suppose $g : A \rightarrow \mathbb{R}$ is bounded; that is, there is a real number $c$ so that $|g(x)| \leq c$ for ...
1
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0answers
24 views

Every uncountable closed set contains a perfect subset

Every uncountable closed set contains a perfect subset. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! My attempt: For $A \...
0
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0answers
28 views

Alternative proofs for Leibniz Rule

Let $f: [a,b] \times [c,d] \to \mathbb{R}$ be a continuous function with $U \subset \mathbb{R}^{n}$ and $\frac{\partial f}{\partial y}$ continuous. Take $F(x,y) = \int_{a}^{b}f(x,y)dx$ and show that ...
0
votes
1answer
26 views

Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$?

Does there exist an open subset $A \subset [0,1]$ such that $m_*(A)\neq m_*(\bar{A})$? I was thinking we could approximate any set from inside by a closed set . This need not true from outside. So ...
1
vote
1answer
13 views

Consider the set $l^\infty$ of all bounded sequences of real numbers. Show that $d$ is well-defined for all pairs of sequences in $l^\infty$

$l^\infty = d(\bf{x},y) = \sup_{n > 0} |x_n - y_n|$ I am having a hard time figuring out how to show it is well-defined. We are to show $d(x,y)$ is always defined. I know that well-defined is ...
1
vote
1answer
28 views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\}.$$ Another way to write this is: ...
1
vote
0answers
40 views

No continuous function $f$ on $\mathbb{R}$ such that $f(x)=1_{[0,1]}(x)$ almost everywhere

Let $1_{[0,1]}$ be the characteristic function of $[0,1]$. Show that there is no everywhere continuous function $f$ on $\mathbb{R}$ such that $$f(x)=1_{[0,1]}(x)\,\,\,\,\,\,\,\,\,\,\,\,\,\text{...
4
votes
0answers
26 views

When does a convex sequence $S_n$ satisfy $S_{a+b+c} - (S_{a+b}+S_{a+c}+S_{b+c})+S_a+S_b+S_c\geq 0$

Let $S_n$ be a sequence of integers with $n\geq 0$ such that $S_0=S_1=S_2=0$ (also assume $S_n$ to not be completely zero) and $$ S_{n-1}+S_{n+1}\geq 2S_n $$ i.e. $S_n$ is convex. Now consider the ...